Jim Burns has brought this to us : > On 10/5/2017 3:12 PM, netzweltler wrote: >> Am Donnerstag, 5. Oktober 2017 17:59:25 UTC+2 >> schrieb Jim Burns: >>> On 10/5/2017 10:00 AM, netzweltler wrote: >>>> Am Donnerstag, 5. Oktober 2017 15:22:35 UTC+2 >>>> schrieb Jim Burns: > >>> [...] >>>>> _We don't do what you're describing_ >>>> >>>> Nevertheless, >>> >>> "Nevertheless"? >>> Do you agree that what you're describing >>> is not what we're doing? > > *NETZWELTLER* > DO YOU AGREE THAT WHAT YOU'RE DOING > IS NOT WHAT WE'RE DOING? > > I think you do agree. > This a pretty fundamental requirement: > When you criticize what someone is doing, > criticize _what they are doing_ and not something else. > >>>> the process >>>> 0 |-> write 0.9 >>>> 1 |-> append another 9 (to the 0.9 already written) >>>> 2 |-> append another 9 (to the 0.99 already written) >>>> ... >>>> results in 0.999... >>>> >>>> Whereas the process you specified earlier >>>> 0 |-> 0.9 >>>> 1 |-> 0.99 >>>> 2 |-> 0.999 >>>> ... >>>> is nothing else but an infinite list of terminating decimals. >>> >>> Right. Nothing else but an infinite list of terminating decimals, >>> which presents no problem, right? >>> >>> And we (meaning _we_ whether or not you include yourself) >>> assign the value of the least upper bound of that list >>> to the non-terminating decimal 0.999... >>> >>> I'm guessing you don't have a problem with the LUB either, >>> because you talk about other things instead. >>> _But this is what we do_ >> >> We obviously agree that the process you specified earlier >> 0 |-> 0.9 >> 1 |-> 0.99 >> 2 |-> 0.999 >> ... >> is nothing else but an infinite list of terminating decimals. > > It think it is also obvious that you have no problem with > an infinite list of terminating decimals. > >> What you don't want to see is, that the process >> 0 |-> write 0.9 >> 1 |-> append another 9 (to the 0.9 already written) >> 2 |-> append another 9 (to the 0.99 already written) >> ... >> results in 0.999... >> >> Maybe you cannot see that I am not writing a new number >> in a new line at each step - as in your process. I am >> appending the 9s in the same line. So I am not creating >> an infinite list of terminating decimals. I am creating >> a single non-terminating decimal. The append operations >> are representing addition operations - infinitely many >> addition operations. > > Suppose, for the sake of argument, that _everything_ that > you have said about what *you* mean by 0.999... is true. > Why does it matter, if it doesn't apply to what *we* mean > by 0.999... ? > > *You* give a meaning to 0.999... that involves infinitely > many addition operations, and then *you* find a problem with > the meaning that *you* gave 0.999... -- a meaning which is > *NOT* the meaning *we* give to 0.999... So what? > > I mean, fine. Whatever. Let me grant, for the sake of argument, > _every error_ that you point out about what *you* mean is > in fact an error. Whoopsie! We'll just have to fix that right > now: We "now" evaluate infinite decimals in a way that avoids > infinite multiplications, whatever they may be. End of problem. > Of course, we already didn't do what your argument suggests > we shouldn't do before you made your argument, but never mind. > _There is no problem_ > > Let me remind you what we're talking about: > > <Burns<netzweltler>> > > > Do you agree that 0.999... means infinitely many > > commands > > Add 0.9 + 0.09 > > Add 0.99 + 0.009 > > Add 0.999 + 0.0009 > > ...? > > 0.999... does not mean infinitely many commands. > > </Burns<netzweltler>>
In fact, I think one could say that the 0.999... decimal expansion representation is the result of having taken the non-terminating decimal expansion representation of, the limit of, the infinte sum as he "builds" above (.9 + .09 +.009 + .0009 ad infinitum) and doing infinitely many truncations thus ending up with the 0.999... notation and the elipsis represents that infinitely many truncation commands had already been done.
That is to say, he might have it exactly backward. The infinite work had to be done to arrive at the particular decimal expansion representation which he wants to start at. The repeating decimal expansion representation was constructed previously and he is trying to deconstruct it by re-appending what was previously truncated.
"The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x."